We now describe the group elements of \(S_{mn}\) . For a detailed derivation of these twist operators see Ref. [1]}. Denoting a specific replica by \((\alpha ,\beta )\) , where \(\alpha = 1,\cdots ,m\)  and \(\beta = 1,\cdots ,n\) , we represent the elements in \(S_{mn}\)  in the following notation:
\(\ell ^{[i]_m} \in S_{mn} \qquad \ell ^{[i]_m}\cdot (\alpha ,\beta ) ={\left\lbrace \begin{array}{ll}(\alpha , \ell \cdot \beta ) & \alpha = i \\(\alpha ,\beta ) & \alpha \ne i\end{array}\right.}\qquad \ell \in S_n\) 
